Mathematical Geosciences

, Volume 49, Issue 2, pp 145–177 | Cite as

A Segmentation Approach for Stochastic Geological Modeling Using Hidden Markov Random Fields

  • Hui Wang
  • J. Florian Wellmann
  • Zhao Li
  • Xiangrong Wang
  • Robert Y. Liang


Stochastic modeling methods and uncertainty quantification are important tools for gaining insight into the geological variability of subsurface structures. Previous attempts at geologic inversion and interpretation can be broadly categorized into geostatistics and process-based modeling. The choice of a suitable modeling technique directly depends on the modeling applications and the available input data. Modern geophysical techniques provide us with regional data sets in two- or three-dimensional spaces with high resolution either directly from sensors or indirectly from geophysical inversion. Existing methods suffer certain drawbacks in producing accurate and precise (with quantified uncertainty) geological models using these data sets. In this work, a stochastic modeling framework is proposed to extract the subsurface heterogeneity from multiple and complementary types of data. Subsurface heterogeneity is considered as the “hidden link” between multiple spatial data sets. Hidden Markov random field models are employed to perform three-dimensional segmentation, which is the representation of the “hidden link”. Finite Gaussian mixture models are adopted to characterize the statistical parameters of multiple data sets. The uncertainties are simulated via a Gibbs sampling process within a Bayesian inference framework. The proposed modeling method is validated and is demonstrated using numerical examples. It is shown that the proposed stochastic modeling framework is a promising tool for three-dimensional segmentation in the field of geological modeling and geophysics.


Geological modeling Geostatistics Uncertainty quantification Gibbs sampling Heterogeneity 



Hui Wang and Florian Wellmann would like to acknowledge the support from the German research foundation (DFG) through the Aachen Institute for Advanced Study in Computational Engineering Science (AICES), RWTH Aachen University. The authors would like to thank the anonymous reviewers for their constructive comments that have helped to improve the paper significantly.


  1. Attias H (2000) A variational Bayesian framework for graphical models. Adv Neural Inf Process Syst 12:209–215Google Scholar
  2. Auerbach S, Schaeben H (1990) Computer-aided geometric design of geologic surfaces and bodies. Math Geol 22:957–987CrossRefGoogle Scholar
  3. Babak O, Deutsch CV (2009) An intrinsic model of coregionalization that solves variance inflation in collocated cokriging. Comput Geosci UK 35:603–614CrossRefGoogle Scholar
  4. Besag J (1974) Spatial interaction and the statistical analysis of lattice systems. J R Stat Soc Ser B (Methodol) 36:192–236Google Scholar
  5. Besag J (1986) On the statistical analysis of dirty pictures. J R Stat Soc 48:259–302Google Scholar
  6. Biernacki C, Celeux G, Govaert G (2000) Assessing a mixture model for clustering with the integrated completed likelihood. IEEE Trans Pattern Anal Mach Intell 22:719–725CrossRefGoogle Scholar
  7. Blanchin R, Chilès J-P (1993) The Channel Tunnel: Geostatistical prediction of the geological conditions and its validation by the reality. Math Geol 25:963–974CrossRefGoogle Scholar
  8. Caers J (2011) Modeling uncertainty in the earth sciences. Wiley, ChichesterCrossRefGoogle Scholar
  9. Caers J, Zhang T (2004) Multiple-point geostatistics: a quantitative vehicle for integrating geologic analogs into multiple reservoir models. G. M. Grammer, P. M. ldquoMitchrdquo Harris, and G. P. Eberli, Integration of outcrop and modern analogs in reservoir modeling. AAPG Memoir 80:383–394Google Scholar
  10. Celeux G, Forbes F, Peyrard N (2003) EM procedures using mean field-like approximations for Markov model-based image segmentation. Pattern Recognit 36:131–144CrossRefGoogle Scholar
  11. Celeux G, Govaert G (1995) Gaussian parsimonious clustering models. Pattern Recognit 28:781–793CrossRefGoogle Scholar
  12. Chugunova TL, Hu LY (2008) Multiple-point simulations constrained by continuous auxiliary data. Math Geosci 40:133–146CrossRefGoogle Scholar
  13. Cline HE, Lorensen WE, Kikinis R, Jolesz F (1990) Three-dimensional segmentation of MR images of the head using probability and connectivity. J Comput Assist Tomogr 14:1037–1045CrossRefGoogle Scholar
  14. Cross GR, Jain AK (1983) Markov random field texture models. IEEE Trans Pattern Anal Mach Intell 5:25–39Google Scholar
  15. Daly C (2005) Higher order models using entropy, Markov random fields and sequential simulation, Geostatistics Banff 2004. Springer, New York, pp 215–224Google Scholar
  16. de Vries LM, Carrera J, Falivene O, Gratacós O, Slooten LJ (2009) Application of multiple point geostatistics to non-stationary images. Math Geosci 41:29–42CrossRefGoogle Scholar
  17. Elkateb T, Chalaturnyk R, Robertson PK (2003) An overview of soil heterogeneity: quantification and implications on geotechnical field problems. Can Geotech J 40:1–15CrossRefGoogle Scholar
  18. Figueiredo MA, Jain AK (2002) Unsupervised learning of finite mixture models. IEEE Trans Pattern Anal Mach Intell 24:381–396CrossRefGoogle Scholar
  19. Fjortoft R, Delignon Y, Pieczynski W, Sigelle M, Tupin F (2003) Unsupervised classification of radar images using hidden Markov chains and hidden Markov random fields. IEEE Trans Geosci Remote Sens 41:675–686CrossRefGoogle Scholar
  20. Forbes F, Peyrard N (2003) Hidden Markov random field model selection criteria based on mean field-like approximations. IEEE Trans Pattern Anal Mach Intell 25:1089–1101CrossRefGoogle Scholar
  21. Fraley C, Raftery AE (1998) How many clusters? Which clustering method? Answers via model-based cluster analysis. Comput J 41:578–588CrossRefGoogle Scholar
  22. Fraley C, Raftery AE (2002) Model-based clustering, discriminant analysis, and density estimation. J Am Stat Assoc 97:611–631CrossRefGoogle Scholar
  23. Gao D (2003) Volume texture extraction for 3D seismic visualization and interpretation. Geophysics 68:1294–1302CrossRefGoogle Scholar
  24. Geman S, Geman D (1984) Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images. IEEE Trans Pattern Anal Mach Intell 6:721–741Google Scholar
  25. Gonzalez J, Low Y, Gretton A, Guestrin C (2011) Parallel Gibbs sampling: From colored fields to thin junction trees, International Conference on Artificial Intelligence and Statistics, pp 324–332Google Scholar
  26. Ising E (1925) Beitrag zur theorie des ferromagnetismus. Zeitschrift für Physik A Hadrons Nuclei 31:253–258Google Scholar
  27. Jessell MW, Valenta RK (1996) Structural geophysics: integrated structural and geophysical modelling. Comput Methods Geosci 15:303–324CrossRefGoogle Scholar
  28. Kindermann R, Snell JL (1980) Markov random fields and their applications. American Mathematical Society, Providence, RIGoogle Scholar
  29. Koch J, He X, Jensen KH, Refsgaard JC (2014) Challenges in conditioning a stochastic geological model of a heterogeneous glacial aquifer to a comprehensive soft data set. Hydrol Earth Syst Sci 18:2907–2923CrossRefGoogle Scholar
  30. Koller D, Friedman N (2009) Probabilistic graphical models: principles and techniques. MIT press, CambridgeGoogle Scholar
  31. Koltermann CE, Gorelick SM (1996) Heterogeneity in sedimentary deposits: A review of structure-imitating, process-imitating, and descriptive approaches. Water Resour Res 32:2617–2658CrossRefGoogle Scholar
  32. Lajaunie C, Courrioux G, Manuel L (1997) Foliation fields and 3D cartography in geology: principles of a method based on potential interpolation. Math Geol 29:571–584CrossRefGoogle Scholar
  33. Li Z, Wang X, Wang H, Liang RY (2016) Quantifying stratigraphic uncertainties by stochastic simulation techniques based on Markov random field. Eng Geol 201:106–122CrossRefGoogle Scholar
  34. Mallet J-L (1989) Discrete smooth interpolation. ACM Trans Gr 8:121–144CrossRefGoogle Scholar
  35. Mallet J-LL (2002) Geomodeling. Oxford University Press Inc, OxfordGoogle Scholar
  36. Mann CJ (1993) Uncertainty in geology. Computers in Geology—25 Years of Progress. Oxford University Press, Oxford, pp 241–254Google Scholar
  37. Mariethoz G, Caers J (2014) Multiple-point geostatistics: stochastic modeling with training images. wiley, New YorkCrossRefGoogle Scholar
  38. Mariethoz G, Renard P, Cornaton F, Jaquet O (2009) Truncated plurigaussian simulations to characterize aquifer heterogeneity. Ground water 47:13–24CrossRefGoogle Scholar
  39. McKenna SA, Poeter EP (1995) Field example of data fusion in site characterization. Water Resour Res 31:3229–3240CrossRefGoogle Scholar
  40. McLachlan G, Peel D (2004) Finite mixture models. Wiley, Hoboken, NJGoogle Scholar
  41. McLachlan GJ, Basford KE (1988) Mixture models. Inference and applications to clustering. Statistics: Textbooks and Monographs. Dekker, New York, p 1Google Scholar
  42. McLachlan GJ, Krishnan T (2007) The EM algorithm and extensions. Wiley-Interscience, New YorkGoogle Scholar
  43. Norberg T, Rosén L, Baran A, Baran S (2002) On modelling discrete geological structures as Markov random fields. Math Geol 34:63–77CrossRefGoogle Scholar
  44. Pham DL, Xu C, Prince JL (2000) Current methods in medical image segmentation. Ann Rev Biomed Eng 2:315–337CrossRefGoogle Scholar
  45. Reitberger J, Schnörr C, Krzystek P, Stilla U (2009) 3D segmentation of single trees exploiting full waveform LIDAR data. ISPRS J Photogramm Remote Sens 64:561–574CrossRefGoogle Scholar
  46. Rubin Y, Chen X, Murakami H, Hahn M (2010) A Bayesian approach for inverse modeling, data assimilation, and conditional simulation of spatial random fields. Water Resour Res 46:W10523CrossRefGoogle Scholar
  47. Rue H, Held L (2005) Gaussian Markov random fields: theory and applications. CRC Press, Boca RatonCrossRefGoogle Scholar
  48. Solberg AHS, Taxt T, Jain AK (1996) A Markov random field model for classification of multisource satellite imagery. IEEE Trans Geosci Remote Sens 34:100–113CrossRefGoogle Scholar
  49. Strebelle S (2002) Conditional simulation of complex geological structures using multiple-point statistics. Math Geol 34:1–21CrossRefGoogle Scholar
  50. Thornton C (1998) Separability is a learner’s best friend, 4th Neural Computation and Psychology Workshop, 9–11 April 1997. Springer, London, pp 40–46Google Scholar
  51. Toftaker H, Tjelmeland H (2013) Construction of binary multi-grid Markov random field prior models from training images. Math Geosci 45:383–409CrossRefGoogle Scholar
  52. Tolpekin VA, Stein A (2009) Quantification of the effects of land-cover-class spectral separability on the accuracy of Markov-random-field-based superresolution mapping. IEEE Trans Geosci Remote Sens 47:3283–3297CrossRefGoogle Scholar
  53. Wang X, Li Z, Wang H, Rong Q, Liang RY (2016) Probabilistic analysis of shield-driven tunnel in multiple strata considering stratigraphic uncertainty. Struct Saf 62:88–100CrossRefGoogle Scholar
  54. Wellmann JF (2013) Information theory for correlation analysis and estimation of uncertainty reduction in maps and models. Entropy 15:1464–1485CrossRefGoogle Scholar
  55. Wellmann JF, Regenauer-Lieb K (2012) Uncertainties have a meaning: Information entropy as a quality measure for 3-D geological models. Tectonophysics 526:207–216CrossRefGoogle Scholar
  56. Wellmann JF, Thiele ST, Lindsay MD, Jessell MW (2016) pynoddy 1.0: an experimental platform for automated 3-D kinematic and potential field modelling. Geosci Model Dev 9:1019–1035CrossRefGoogle Scholar
  57. Xie H, Pierce LE, Ulaby FT (2002) SAR speckle reduction using wavelet denoising and Markov random field modeling. IEEE Trans Geosci Remote Sens 40:2196–2212CrossRefGoogle Scholar
  58. Yuen KV, Mu HQ (2011) Peak ground acceleration estimation by linear and nonlinear models with reduced order Monte Carlo simulation. Comput Aided Civil Infrastruct Eng 26:30–47Google Scholar
  59. Zhang Y, Brady M, Smith S (2001) Segmentation of brain MR images through a hidden Markov random field model and the expectation-maximization algorithm. IEEE Trans Med Imaging 20:45–57CrossRefGoogle Scholar
  60. Zhu H, Zhang L (2013) Characterizing geotechnical anisotropic spatial variations using random field theory. Can Geotech J 50:723–734CrossRefGoogle Scholar

Copyright information

© International Association for Mathematical Geosciences 2016

Authors and Affiliations

  • Hui Wang
    • 1
  • J. Florian Wellmann
    • 1
  • Zhao Li
    • 2
  • Xiangrong Wang
    • 3
  • Robert Y. Liang
    • 2
  1. 1.The Aachen Institute for Advanced Study in Computational Engineering Science (AICES)RWTH Aachen UniversityAachenGermany
  2. 2.Department of Civil EngineeringThe University of AkronAkronUSA
  3. 3.College of EngineeringPeking UniversityBeijingChina

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